Question: A circle with area $81\pi$ has a sector with a $40^\circ$ central angle. What is the area of the sector? ${81\pi}$ $\color{#9D38BD}{40^\circ}$ ${9\pi}$
Solution: The ratio between the sector's central angle $\theta$ and $360^\circ$ is equal to the ratio between the sector's area, $A_s$ , and the whole circle's area, $A_c$ $\dfrac{\theta}{360^\circ} = \dfrac{A_s}{A_c}$ $\dfrac{40^\circ}{360^\circ} = \dfrac{A_s}{81\pi}$ $\dfrac{1}{9} = \dfrac{A_s}{81\pi}$ $\dfrac{1}{9} \times 81\pi = A_s$ $9\pi = A_s$